Weakly left localizable rings. Issue 9 (2nd September 2017)

• Record Type: Journal Article
• Authors: Bavula, V. V.
• Is Part Of: Communications in algebra. Volume 45:Issue 9(2017)
• Page Start: 3798
• Page End: 3815
• Publication Date: 2017-09-02
• Subjects: A left localization maximal ring -- a weakly left localizable ring -- denominator set -- the classical left quotient ring of a ring -- the largest left quotient ring of a ring -- the largest regular left Ore set of a ring; 16U20 -- 16P50 -- 16S85; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1247451
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: A new class of rings, the class of weakly left localizable rings, is introduced. A ring R is called weakly left localizable if each non-nilpotent element of R is invertible in some left localization S −1 R of the ring R . Explicit criteria are given for a ring to be a weakly left localizable ring provided the ring has only finitely many maximal left denominator sets (eg, this is the case for all left Noetherian rings). It is proved that a ring with finitely many maximal left denominator sets that satisfies some natural conditions is a weakly left localizable ring iff its left quotient ring is a direct product of finitely many local rings such that their radicals are nil ideals.